In compensated elemental metals at low temperature, a several Tesla field can suppress electronic heat conduction so thoroughly that heat is effectively carried by phonons alone. In approximately one mm diameter single crystal samples with impurity concentrations low enough that electron conduction is limited by surface scattering, the ratio of zerofield to high-field thermal conductivity can exceed ten thousand. We have used this phenomenon to build a compact, solid-state heat switch with no moving parts and no enclosed fluids. The time scale for switching states is limited by time scale for charging the magnet that supplies the controlling field. Our design and fabrication techniques overcome the difficulties associated with manufacturing and assembling parts from single crystal tungsten. A clear disadvantage of the magnetoresistive switch is the mass and complexity of the magnet system for the controlling field. We have discovered a technique of minimizing this mass and complexity, applicable to the continuous adiabatic demagnetization refrigerator.